Laplace filter

The Laplace filter or discrete Laplace operator is a filter for edge detection that approximates the Laplace operator (sum of the two pure second derivatives):

{\ displaystyle \ Delta f = {\ frac {\ partial ^ {2} f} {\ partial x ^ {2}}} + {\ frac {\ partial ^ {2} f} {\ partial y ^ {2} }}}

An edge is understood to be a curve along which the gradient of the image always points in the normal direction. The vector field is therefore free of sources in the area of the edge. An edge can therefore only appear if the following equation is fulfilled: ${\ displaystyle \ nabla f}$

{\ displaystyle 0 = \ nabla \ cdot (\ nabla f) = \ Delta f}

So you are looking for the zero crossings of a Laplace-filtered image. However, it should be noted here that even homogeneous areas are equal to zero. The Laplace filter therefore only provides a superset of the possible edges. ${\ displaystyle \ Delta f}$

functionality

Calculation of the second derivative of edges in a noisy 1D signal

The figure on the right shows a noisy signal from which the second derivative was calculated. The edge appears here as the zero crossing of the signal. The Laplace operator is applied to a discrete signal g _n or g _nm via a convolution . You can use the following simple convolution masks :

1D:

{\ displaystyle \ quad {\ vec {D}} _ {x} ^ {2} \; = {\ begin {bmatrix} 1 & -2 & 1 \ end {bmatrix}}}

2D:

{\ displaystyle \ quad \ mathbf {D} _ {xy} ^ {2} = {\ begin {bmatrix} 0 & 1 & 0 \\ 1 & -4 & 1 \\ 0 & 1 & 0 \ end {bmatrix}}}

There is a second variant for the 2D filter, which, in contrast to the variant above, also responds to 45 ° edges:

2D:

{\ displaystyle \ quad \ mathbf {D} _ {xy} ^ {2} = {\ begin {bmatrix} 1 & 1 & 1 \\ 1 & -8 & 1 \\ 1 & 1 & 1 \ end {bmatrix}}}

These convolution masks are obtained by discretizing the difference quotients. At the end of the article you can see examples of the application of the Laplace filter.

Transfer function and isotropy of the filter

Transfer function of various Laplace filters

The transfer function ( Fourier transform ) of the ideal Laplace operator Δ is:

{\ displaystyle \ Delta = {\ frac {\ partial ^ {2}} {\ partial x ^ {2}}} + {\ frac {\ partial ^ {2}} {\ partial y ^ {2}}} \ \ \ \ \ circ - \ bullet \ \ \ \ -4 \ pi ^ {2} | {\ vec {k}} | ^ {2}}

A discretized Laplace operator must approximate this parabolic transfer function as well as possible.

The figure on the right shows the transfer function of the first 2D Laplace filter. You can clearly see the anisotropy and the high-pass character of the transfer function. The formula is:

{\ displaystyle {\ hat {l}} ({\ vec {k}}) = - 4 \ sin ^ {2} (\ pi k_ {x} / 2) -4 \ sin ^ {2} (\ pi k_ {y} / 2)}

It shows similarity to the ideal transfer function of the Laplace operator. ${\ displaystyle {\ vec {k}} = 0}$

One arrives at an isotropic approximation of the Laplace operator if one chooses a somewhat different representation of the Laplace filter:

{\ displaystyle 4 ({\ mathcal {B}} ^ {2} - {\ mathcal {I}}) = {\ frac {1} {4}} {\ begin {bmatrix} 1 & 2 & 1 \\ 2 & -12 & 2 \\ 1 & 2 & 1 \ end {bmatrix}}}

There is the 3 × 3 binomial filter (smoothing filter ) and a "unit filter" / δ-pulse that maps the image onto itself (the point response is zero everywhere, except for the central pixel. There it is 1). The transfer function of this filter is: ${\ displaystyle {\ mathcal {B}} ^ {2}}$ ${\ displaystyle {\ mathcal {I}}}$

{\ displaystyle {\ hat {l}} ({\ vec {k}}) = - 4 \ cos ^ {2} (\ pi k_ {x} / 2) \ cdot \ cos ^ {2} (\ pi k_ {y} / 2) -4}

This transfer function is also included in the figure on the right. It turns out that it is much more isotropic than the first version.

example Pictures

Original image
Image filtered with the simplest filter mask (click to enlarge)
Image filtered with the second filter mask (click to enlarge)

software

The Laplace filter can be executed with the graphics program GIMP via the menu calls Filter -> Find edges -> Edges. It is also implemented in the free image processing libraries Scikit-image and OpenCV .

literature

Bernd Jähne : Digital image processing. 6th, revised and expanded edition. Springer-Verlag, Berlin 2005, ISBN 3-540-24999-0
Russell Merris: Laplacian matrices of graphs: a survey. In: Linear Algebra and its Applications. 197-198, 143-176 (1994). ISSN 0024-3795

Individual evidence

↑ 7.4. Laplace. In: GNU Image Manipulation Program - User Guide. GIMP, accessed November 29, 2018 .
↑ Module: filters - skimage v0.15.dev0 docs. Retrieved November 29, 2018 .
^ OpenCV: Laplace Operator. Retrieved November 29, 2018 .

[1] 7.4. Laplace. In: GNU Image Manipulation Program - User Guide. GIMP, accessed November 29, 2018 .

[2] Module: filters - skimage v0.15.dev0 docs. Retrieved November 29, 2018 .

[3] OpenCV: Laplace Operator. Retrieved November 29, 2018 .